CN106935033A - The ofaiterative, dynamic linearisation of Expressway Traffic system and Learning Control Method - Google Patents

Abstract

The present invention relates to Expressway Traffic control technology field, and in particular to a kind of ofaiterative, dynamic linearisation of Expressway Traffic system and Learning Control Method, comprise the following steps：(1) the spatial spreading traffic model of Expressway Traffic system is set up；(2) form of the spatial spreading traffic flow model General Nonlinear Discrete_Time System is represented；(3) general nonlinearity discrete time model is converted into dynamical linearization data model；(4) the study control law and parameter more new law of dynamical linearization data model are set up.LDM AILC methods proposed by the invention can process nonlinear system, and without the structure of known linear parameter, it is a kind of control method of data-driven, and the design and analysis of controller are solely dependent upon I/O data.Additionally, under the conditions of the non-critical with random initial state and iteration change tracking target is repeatable, the LDM AILC methods for being proposed remain able to obtain perfect tracking performance.

Description

The ofaiterative, dynamic linearisation of Expressway Traffic system and Learning Control Method

Technical field

The present invention relates to Expressway Traffic control technology field, and in particular to a kind of ofaiterative, dynamic of Expressway Traffic system Linearisation and Learning Control Method.

Background technology

Expressway Traffic control is a key areas in traffic engineering and intelligent transportation system.Peak period highway Frequent congestion deteriorate traffic.Causing the most common reason of through street congestion includes：Transport need more than designed capacity, Traffic accident, road engineering and weather conditions.In order to preferably play the performance of through street, Entrance ramp is the plan for generally using Slightly.The purpose of On-ramp Control is to adjust the volume of traffic at its Entrance ramp into through street main stem, it is ensured that under Keep expecting the vehicle flowrate of (or optimal) on trip main line through street so that Quick Traffic Capacity reaches maximization.In practice, may be used At Entrance ramp, statistic mixed-state and control are carried out to the automobile quantity for entering by Traffic monitoring device and signal timing dial device System.

At present, Expressway Traffic system has been successfully applied to based on ALINEA Entrance ramp strategies that are local and coordinating to feed back In control.It has proved to be a kind of very simple, the On-ramp Control method realized efficiently and conveniently.However, the method The key model parameter of calibration influence systematic function is difficult to, because model parameter can be with the geometry of road conditions and environment Factor, such as rainwater or greasy weather and it is different.Additionally, Expressway Traffic streaming system is strong nonlinearity, couples and uncertain, Therefore its accurate model is hardly resulted in practice.So, common System design based on model method is applied to through street During traffic control problems, many unexpected difficulties can be often run into.

On the other hand, macro-traffic stream mode is typically all daily repetition.For example, traffic flow will since midnight and by Cumulative to add to first peak i.e. morning peak, generally from 7 points to 9 points of morning, second peak is from 5 points to 7 points of afternoon.Congestion is led to Often start in same position daily.Iterative learning control (ILC) is especially suitable for processing the control problem of repetitive process.In early stage In research, there is scholar to propose some and be based on the ILC methods of Entrance ramp strategy, and be applied to through street one day or one week Density domination in.In document " Freeway traffic control using iterative learning control- In based ramp metering and speed signaling ", Entrance ramp and speed control based on ILC are discussed. In document " An iterative learning approach for density control of freeway traffic In flow via ramp metering ", the study mechanism being combined with pure error feedback in complementary fashion, emulation knot are have studied Fruit demonstrates the superiority based on ILC.Document " A complementary modularized ramp metering Approach based on iterative learning control and ALINEA ", by using entering based on ILC Mouthful ring road strategy and the complementary Entrance ramp strategy that is combined based on ILC and ALINEA are studied input saturation.In text Offer " Iterative learning control of freeway flow via ramp metering and Simulation on PARAMICS " and document " Flow based local ramp metering using iterative In learning and PARAMICS platform ", the validity of the On-ramp Control based on ILC is further have evaluated.

It should be noted that the above-mentioned ILC methods for On-ramp Control are all based on compression mapping and fixing point reason By the linear iteraction learning algorithm of design, can there are two main limitations in this in actual applications.First limitation be, due to The convergence of tracking error is obtained based on λ norms, is deteriorated along the transient response performance that iteration axle system is exported sometimes.Second Individual limitation is that identical original state and same reference track must match and could obtain perfect tracking.

Strict repeatability day by day is a kind of preferable traffic control situation.In practice, due to the road of through street Conditions and environment factor, such as rainwater and greasy weather so that the initial value of traffic current density and vehicle average speed, and tracking mesh Mark is all daily change.Therefore, the expectation with Random Initial Condition and iteration change of research non-critical repeat pattern is close The ILC methods for spending the Expressway Traffic system of track are with definite meaning.

At present, adaptive iterative learning control (AILC) scheme has obtained widely studied, and has many successful stories. In general, the most attractive advantages of AILC are the abilities of the reference locus of its treatment iteration change, and these Reference locus are bounded but may have the problems such as big random initially reset error and interference.However, most of existing AILC depends on such a fact, i.e. unknown parameter by known nonlinear function linear parameterization, due to Macro-traffic Flow mould Type has a strong nonlinearity, therefore cannot be directly used to Expressway Traffic control problem.

Recently, neutral net or fuzzy system have been applied in AILC, cannot linearly be joined with solving nonlinear system The problem of numberization.It is however typically difficult to determine fuzzy rule base and membership function, and god is trained with substantial amounts of peration data It is also relatively difficult through network.Recently, scholar is incorporated into iteration domain by by dynamical linearization method, it is proposed that for general non- A kind of model-free adaption ILC of the data-driven of linear discrete time system.It realizes complete under Random Initial Condition Tracking, and do not need any external test signal or training process.But, target following track must be strict repetition.

Inspired by discussed above, by introducing the new ideas of pseudo- partial derivative (PPD), proposed for Expressway Traffic system A kind of new dynamical linearization method of equal value.Further, it is proposed that a kind of ofaiterative, dynamic based on Expressway Traffic system The Learning Control Method (Linear Data-model based Adaptive ILC, LDM-AILC) of linearisation, by circle Road regulation is controlled to Expressway Traffic current density.

The content of the invention

The purpose of the present invention is to propose to the ofaiterative, dynamic linearisation and self study control of a kind of new Expressway Traffic system Method processed, is controlled by ramp metering to Expressway Traffic current density.

To achieve these goals, the present invention adopts the following technical scheme that the ofaiterative, dynamic of Expressway Traffic system is linear Change and Learning Control Method, comprise the following steps：

(1) the spatial spreading traffic model of Expressway Traffic system is set up；

(2) form of the spatial spreading traffic flow model General Nonlinear Discrete_Time System is represented；

(3) general nonlinearity discrete time model is converted into dynamical linearization data model；

(4) the study control law and parameter more new law of dynamical linearization data model are set up.

Further, Expressway Traffic system postulation described in the step (1) includes an one-lane through street, often One section has an Entrance ramp and one outlet ring road, then the spatial spreading traffic flow model of the through street is：

q_i(t)=ρ_i(t)v_i(t), (2)

Wherein h is sampling time interval；T refers to t-th moment；i∈{1,Λ,I_NRefer to through street i-th section； I_NIt is total sector number；τ, υ, κ, l, m are constant parameter；ρ_iT () represents that the traffic flow at i-th of through street, t-th moment of section is close Degree；v_iT () represents i-th of through street, t-th of the section average speed at moment；q_iWhen () represents i-th section t-th in through street t The magnitude of traffic flow at quarter；r_iT () represents the Entrance ramp traffic flow rate at i-th of through street, t-th of section moment；s_iT () represents quick The exit ramp traffic flow rate at i-th of road, t-th of section moment；L_iRepresent the length of i-th of through street section, V_freeRepresent fast The rubato of i-th of fast road section, ρ_jamRepresent maximal density.

Further, the spatial spreading traffic flow model is converted into general nonlinearity form is：

Y (t+1)=f [y (t), r (t), d (t)], (5)

Wherein, state vector y (t) ∈ RⁿIncluding all traffic densities, average speed and ring road sequence；Dominant vector r (t)∈RⁿIncluding all controllable ring road flow rates；Interference vector d (t) ∈ RⁿDemand and turning speed including all Entrance ramps；f (Λ)∈RⁿIt is vector valued function.

Further, general nonlinearity discrete time model is converted into dynamical linearization data mould in the step (3) Type meets following 2 hypothesis, it is necessary to set nonlinear data model：

Assuming that 1：The partial derivative of f (Λ) on dominant vector r (t) is continuous；

Assuming that 2：Nonlinear data model meets generalized Lipschitz condition, i.e., to any fixed t and | | Δ r (t) | | ≠ 0, have

Wherein, Δ y (t+1)=y (t+1)-y (t), Δ r (t+1)=r (t)-r (t-1)；It is a normal number；

Can then obtain, a parameter for being referred to as PPD matrixes is certainly existed for arbitrary moment tSo that non-linear number The dynamical linearization data model of following equivalence can be converted into according to model,

Δ y (t+1)=Φ (t) Δs r (t) (6)

Wherein, Φ (t) ∈ R^n×nAnd | | Φ (t) | |≤b_Φ。

Further, the study control law and parameter more new law of dynamical linearization data model are set up in the step (5) The step of be：

(51) set dynamical linearization data model and meet hypothesis 3, concurrently set Expressway Traffic system in limited operation Repeated in time interval t,It is strict repetition；

Assuming that 3：PPD parameter matrixsIt is positive definite or nonnegative definite；

(52) set expectation traffic and be output as y_d,k(t)∈Rⁿ, for all of t ∈ { 0,1, ∧, T }, k=1,2, ∧, y_d,k T () is that iteration is related and bounded, i.e.,

Wherein, b_ydIt is normal number and presence；

(53) tracking error e is defined_k(t)=y_d,k(t)-y_k(t), then

e_k(t+1)=y_d,k(t+1)-y_k(t)-Φ(t)Δr_k(t)=Φ (t) (Φ (t)^-1y_d,k(t+1)-Φ(t)^-1y_k (t)-Δr_k(t)) (9)

Order

e_k(t+1)=Φ (t) [Θ (t) ζ_k(t)-Δr_k(t)] (10)

Wherein, ζ_k(t)=y_d,k(t+1)-y_k(t)∈Rⁿ, Θ (t)=Φ (t)^-1∈R^n×n；

(54) the study control law that can then obtain kth time can be expressed as：

Wherein,It is the estimate of Θ (t).Its parameter more new law is

Wherein,It is given bounded；C ＞ 0；0 ＜ ab_Φ＜ 2, P=I_n×nIt is unit matrix.

LDM-AILC methods proposed by the invention can process nonlinear system, and without the knot of known linear parameter Structure, it is a kind of control method of data-driven, and the design and analysis of controller are solely dependent upon I/O data.Additionally, with Under the conditions of the non-critical of machine original state and iteration change tracking target is repeatable, the LDM-AILC methods for being proposed are remained able to Obtain perfect tracking performance.Therefore, in practice more suitable for the repeatable condition of higher order, strong nonlinearity and non-critical This typical complicated Expressway Traffic control system.Theory analysis and simulation result confirm the validity of institute's extracting method. And the method have the advantages that：

(1) the ofaiterative, dynamic linearization technique of Expressway Traffic system of the invention, compared with former method, it is not necessary to Model and equivalent；

(2) method of the present invention, based on data-driven, is to update current operation using the former information for repeating, Empirical learning with people is similar；

(3) method of the present invention need not require that system brings into operation from same initial point daily；

(4) method of the present invention is when daily expected density and desired speed are changed, it is also possible to apply.

Brief description of the drawings

Fig. 1 is each section description in through street with inlet/outlet ring road；

Fig. 2 is the expectation traffic density distribution map of iteration change；

Fig. 3 is the variation diagram of initial traffic density iteration 100 times；

Fig. 4 is the maximum tracking error variation diagram in time interval t ∈ { 0, Λ, 500 }.

Specific embodiment

The present invention is further illustrated with reference to the accompanying drawings and examples.

As shown in figure 1, Expressway Traffic system includes an one-lane through street, each section has an entrance circle Road and one outlet ring road.Shown in following (1)-(4) formula of its spatial spreading traffic flow model.

q_i(t)=ρ_i(t)v_i(t), (2)

Wherein, h is sampling time interval；T refers to t-th moment, t ∈ { 0,1, ∧, T }；i∈{1,∧,I_NRefer to fast I-th section on fast road；I_NIt is total sector number；τ, v, k, l, m are constant parameter；ρ_iT () represents i-th of through street section t The traffic current density at individual moment；v_iT () represents i-th of through street, t-th of the section average speed at moment；q_iT () represents through street I-th section, t-th magnitude of traffic flow at moment；r_iT () represents the Entrance ramp traffic at i-th of through street, t-th of section moment Flow rate；s_iT () represents the exit ramp traffic flow rate at i-th of through street, t-th of section moment；L_iRepresent i-th of through street section Length, V_freeRepresent the rubato of i-th of through street section, ρ_jamRepresent maximal density.

Assuming that Expressway Traffic system reruns in limited time interval t={ 0,1, Λ, T }.Control targe is Design it is a kind of need not known definite traffic flow model and disturbed condition self adaptation ILC methods.Self adaptation ILC will be using going through The Expressway Traffic data of history produce control input sequence, to cause traffic density on whole interval t={ 0,1, Λ, T } Converge to desired value.

According to (1)-(4) formula, the spatial spreading traffic flow model general nonlinearity discrete-time version is expressed as：

Y (t+1)=f [y (t), r (t), d (t)], (5)

Wherein, state vector y (t) ∈ RⁿComprising all traffic densities, average speed and ring road sequence；Dominant vector r (t)∈RⁿIncluding all controllable ring road flow rates；Interference vector d (t) ∈ R^pDemand and turning speed comprising all Entrance ramps；f (Λ)∈RⁿIt is vector valued function.

Assuming that 1：The partial derivative of f (Λ) on control input r (t) is continuous.

Assuming that 2：Nonlinear data model meets generalized Lipschitz condition, i.e., to any fixed t and | | Δ r (t) | | ≠ 0, have

||Δy(t+1)||≤b_Φ||Δr(t)||

Wherein, Δ y (t+1)=y (t+1)-y (t), Δ r (t)=r (t)-r (t-1)；b_ΦIt is a normal number.

Assuming that 1 is the representative condition of General Nonlinear Systems controller design.Assuming that 2 limitations are driven by the change of control input The rate of change of dynamic system output, it means that the limited change of the Entrance ramp magnitude of traffic flow will not cause the unlimited of traffic density Change.Additionally, we only need to known b_ΦThe presence of such a constant, without known its exact value.

The general nonlinearity discrete time model of 1 and hypothesis 2 is assumed for meeting, when | | Δ r (t) | | is when ≠ 0, for appointing The moment t of meaning certainly exists parameter Φ (t) for being referred to as PPD matrixes so that nonlinear data model can change into as Lower dynamical linearization data model of equal value,

Δ y (t+1)=Φ (t) Δs r (t) (6)

Wherein, Φ (t) ∈ R^n×nAnd | | Φ (t) | |≤b_Φ。

By nonlinear data model,

Make Ψ (t)=f [y (t), r (t-1), d (t)]-f [y (t-1), r (t-1), d (t-1)].By in hypothesis 1 and differential Value theorem, (A1) is rewritable to be

Wherein, Represent f_iOn input r_j(t) In interval [r_j(t),r_j(t-1) the local derviation numerical value in] at certain point.For the t that each is fixed, it is contemplated that below equation, H (t) For the numerical matrix that n rows, n are arranged：

Ψ (t)=H (t) Δs r (t) (A3)

Because condition, | | Δ r (t) | | ≠ 0 meets, at least one solution of equation (A3) H^*(t).In fact, during for each T is carved, it necessarily has infinite multiple solutions.

OrderSo we have Δ x (t+1)=Φ (t) Δs r (t).As a result | | Φ (t) | |≤b_ΦAssume that 2 direct conclusion.

Dynamical linearization data model is the description a kind of of equal value to general nonlinearity discrete time model, it and other Linear forms it is different, such as Taylor linearization eliminates higher order term.Dynamical linearization is a kind of method of data-driven, its reality Now only depend on the inputoutput data of system.Additionally, linear data model is very simple, it is not necessary to any fuzzy control rule Then, external test signal and the training process as neutral net.

Another hypothesis is on PPD parameters.

PPD parameter matrixs Φ (t) is positive definite or nonnegative definite.Without loss of generality, herein we assume that Φ (t) >=δ I ＞ 0.

Assuming that 3 define the same tropism of control direction, this is in the controls common.

Such as document " Data driven model-free adaptive control for a class of MIMO Nonlinear discrete-time systems " are described, and Φ (t) represents the transmission letter of Markovian parameter or linear system Number, for the Expressway Traffic system for only changing in limited time interval t ∈ { 0,1, Λ, T } rise times axle, can be reasonably Assuming that Φ (t) is strict repeatable, then by considering the repeatability of Expressway Traffic control system, dynamical linearization data The equivalent expression of model can be expressed as,

y_k(t+1)=y_k(t)+Φ(t)Δr_k(t) (10)

Wherein, Δ r_k(t)=r_k(t)-r_k(t-1)；T={ 0,1, Λ, T }；K=1,2, Λ represent iterations.

Desired traffic output is y_d,k(t)∈Rⁿ, for all of t ∈ { 0, Λ, T }, k=1,2, Λ, it is iteration phase Close and bounded, i.e.,

Wherein, we only need to know normal number b_ydExistence.

Define tracking error e_k(t)=y_d,k(t)-y_k(t),t∈{0,1,Λ,T}.By (10) formula, we can obtain

e_k(t+1)=y_d,k(t+1)-y_k(t)-Φ(t)Δr_k(t)=Φ (t) (Φ (t)^-1y_d,k(t+1)-Φ(t)^-1y_k (t)-Δr_k(t)) (11)

Make ζ_k(t)=y_d,k(t+1)-y_k(t)∈Rⁿ, Θ (t)=Φ (t)^-1∈R^n×n.Equation (11) it is rewritable into

e_k(t+1)=Φ (t) [Θ (t) ζ_k(t)-Δr_k(t)] (12)

So, the study control law of kth time can be expressed as follows

Wherein,It is the estimate of Θ (t).Its parameter more new law is

Wherein,It is given bounded；C ＞ 0；0 ＜ ab_Φ＜ 2, b_ΦIt is normal number as assumed defined in 2；P= I_n×nIt is unit matrix.

Wherein, with other self adaptation ILC^[14-20]Difference, ζ_k(t)=y_d,k(t+1)-y_kT () is linear function, with system Output is relevant with reference locus.Therefore, meet on linear growth condition automatically.

Wherein, the LDM-AILC for being carried is a kind of method of data-driven, because the design of controller and analysis are only used The data surveyed of system input and output.It, with iterative estimate, is also only to use the measurable I/ of control system that unknown parameter Θ (t) is O data.

For the LDM-AILC schemes that MIMO nonlinear discrete time systems are proposed, assuming that 1-3 meet under conditions of, Control law (13) ensure that with study more new law (14)：

A () is for all of t ∈ { 0, Λ, T }, k=1,2, Λ, PPD Matrix EstimationIt is bounded.

(b) when k tends to infinite, tracking error e_kT (), t ∈ { 1, Λ, T }, goes to zero along iteration axle.

Defined parameters evaluated errorControl law (13) formula is substituted into error dynamics equation (12) formula In, obtain

Note following property

Property 1.

Property 2.trace (Q^Tvy^T)=trace (Q^Tvy^T)^T=v^TQy

Wherein, A, B, C are square formation, Q ∈ R^m×n, v ∈ R^m×1, y ∈ R^n×1。

DefinitionSo according to above property 1, can obtain

According to (14) formula, equation (B2) becomes

Due to above property 2, (B3) is changed into

Θ (t) is subtracted simultaneously on (14) formula both sides, and with relational expression (B1), can be obtained

In view of (B5), (B4) formula is rewritable to be

Relational expression (B1), Wo Menyou are used again

Due to 0 ＜ ab_Φ＜ 2, q ＞ 0, it is evident that

By (B7) and (B8), it is easy to obtain

Or

According to theorem 1 and hypothesis 3,0 ＜ δ≤| | Φ (t) | |≤b_Φ, so Θ (t) is bounded.Further,t ∈ { 0, Λ, T }, is given bounded, it is therefore apparent thatIt is bounded.Can be released by inequality (B10) againIt is non-negative, non-increasing and bounded, soIt is bounded.

The both sides of (B7) are sued for peace from 0 to k, is obtained

Due to V₀T () is bounded, V_kT () is non-to bear boundary, it is contemplated that (B8) and (B11), can be obtained

According to ζ_kT the definition of (), has

Wherein,q₂=1 is two normal numbers.

Therefore, it is that can obtain for all of t ∈ { 1, Λ, T }, e according to convergence (B12) and (B13)_kT () edge changes For axle asymptotic convergence.

In order to verify the correctness of the inventive method, following emulation is carried out to the method for the present invention：

For emulation, it is contemplated that the through street in an interval is divided into 12 sections.The length of each section is 0.5km.The initial volume of traffic into the 1st section is 1600 vehicles per hour.The parameter used in the model is as follows：v_i(0) =50km/h, v_free=80km/h, ρ_jam=80veh/lane/km, l=1.8, m=1.7, κ=13veh/km, τ=0.01h, h =0.00417h, γ=35km²/ h, r_i(0)=0veh/h, α=0.95.

There is an Entrance ramp mouthful in the 2nd section, it is known that transport need, there are two exit ramp mouths to be located at the 5th area respectively Section and the 9th section, rate of discharge are unknown.The traffic conditions of peak period are simulated with this.Unknown rate of discharge is actually As the external disturbance of the 2nd section of simulation.

Note, sequence requirement is actually applied with some constraints to the control input of Entrance ramp, for example：It is carved into k The traffic flow rate of mouthful ring road no more than current demand with currently Entrance ramp wait sequence and；Therefore

Wherein, l_iT () refers to the length in i-th Entrance ramp of t wait sequence that may be present；η_iT () is in t I-th Entrance ramp transport need amount (veh/h) of moment；The I in emulating herein_ON=2, refer to there is the sector number of Entrance ramp. On the other hand, accumulation of the demand with the difference of actual flow that sequence is Entrance ramp is waited, i.e.,

l_i(t+1)=l_i(t)+T[η_i(t)-r_i(t)], i ∈ I_ON (16)

Desired Expressway Traffic density is ρ_d,k=30+0.1sin (π k/50), as shown in Fig. 2 it is with iterations It is continually changing.Random initial traffic density choosing is ρ_i,k(0)=30+0.01rand, as shown in Figure 3.

In simulations, we select a=0.1, c=0.01, θ₀(t)=0.002, u₀(t)=0.Using the LDM- for being carried AILC methods, Learning Convergence is as shown in Figure 4.Wherein transverse axis is iterations, and the longitudinal axis is the maximum value of tracking error

The validity of put forward LDM-AILC methods is can be seen that from Fig. 2-4.Although initial value is random and reference locus edge changes For axle change at random, tracking error still progressively approaches zero.

Claims (5)

1. the ofaiterative, dynamic of Expressway Traffic system is linearized and Learning Control Method, it is characterised in that comprised the following steps：

(1) the spatial spreading traffic model of Expressway Traffic system is set up；

(2) form of the spatial spreading traffic flow model General Nonlinear Discrete_Time System is represented；

(3) general nonlinearity discrete time model is converted into dynamical linearization data model；

(4) the study control law and parameter more new law of ofaiterative, dynamic linearized data model are set up.

2. the ofaiterative, dynamic of Expressway Traffic system according to claim 1 is linearized and Learning Control Method, and it is special Levy and be, Expressway Traffic system described in the step (1) includes an one-lane through street, each section has one Entrance ramp and one outlet ring road, then the spatial spreading traffic flow model of the through street be：

${\mathrm{\ρ}}_i(t+1)={\mathrm{\ρ}}_i\left(t\right)+\frac{h}{L_i}\[q_{i-1}\left(t\right)-q_i\left(t\right)+r_i\left(t\right)-s_i\left(t\right)\],---\left(1\right)$

q_i(t)=ρ_i(t)v_i(t), (2)

$v_i(t+1)=v_i\left(t\right)+\frac{h}{\mathrm{\τ}}\[V\left({\mathrm{\ρ}}_i\right(t\left)\right)-v_i\left(t\right)\]+\frac{h}{L_i}{v}_i\left(t\right)\[v_{i-1}\left(t\right)-v_i\left(t\right)\]-\frac{\mathrm{\υ}h}{{\mathrm{\τL}}_i}\frac{\[{\mathrm{\ρ}}_{i+1}\left(t\right)-{\mathrm{\ρ}}_i\left(t\right)\]}{\[{\mathrm{\ρ}}_i\left(t\right)+\mathrm{\κ}\]},---\left(3\right)$

$V\left({\mathrm{\ρ}}_i\right(t\left)\right)=v_{free}{(1-{\[\frac{{\mathrm{\ρ}}_i\left(t\right)}{{\mathrm{\ρ}}_{jam}}\]}^l)}^m,---\left(4\right)$

Wherein, h is sampling time interval；T refers to t-th moment, t ∈ { 0,1, ∧, T }；i∈{1,∧,I_NIt refer to through street I-th section；I_NIt is total sector number；τ, v, k, l, m are constant parameter；ρ_iT () represents i-th of through street, t-th moment of section Traffic current density；v_iT () represents i-th of through street, t-th of the section average speed at moment；q_iT () represents through street i-th T-th magnitude of traffic flow at moment of section；r_iT () represents the Entrance ramp traffic flow rate at i-th of through street, t-th of section moment；s_i T () represents the exit ramp traffic flow rate at i-th of through street, t-th of section moment；L_iRepresent the length of i-th of through street section Degree, V_freeRepresent the rubato of i-th of through street section, ρ_jamRepresent maximal density.

3. the ofaiterative, dynamic of Expressway Traffic system according to claim 2 is linearized and Learning Control Method, and it is special Levy and be, the spatial spreading traffic flow model is converted into general nonlinearity discrete-time version is：

Y (t+1)=f [y (t), r (t), d (t)], (5)

Wherein, state vector y (t) ∈ RⁿIncluding all traffic densities, average speed and ring road sequence；Dominant vector r (t) ∈ RⁿIncluding all controllable ring road flow rates；Interference vector d (t) ∈ RⁿDemand and turning speed including all Entrance ramps；f(Λ) ∈RⁿIt is vector valued function.

4. the ofaiterative, dynamic of Expressway Traffic system according to claim 3 is linearized and Learning Control Method, and it is special Levy and be, in the step (3) by nonlinear data model conversation be dynamical linearization data model, it is necessary to set non-linear number Meet following 2 hypothesis according to model：

Assuming that 1：The partial derivative of f (Λ) on dominant vector r (t) is continuous；

Assuming that 2：Nonlinear data model meets generalized Lipschitz condition, i.e., to any fixed t and ‖ Δs r (t) ‖ ≠ 0, have

Wherein, Δ y (t+1)=y (t+1)-y (t), Δ r (t+1)=r (t)-r (t-1)；It is a normal number；

Can then obtain, a parameter for being referred to as PPD matrixes is certainly existed for arbitrary moment tSo that nonlinear data mould Type can be converted into the dynamical linearization data model of following equivalence,

Wherein,And

5. the ofaiterative, dynamic of Expressway Traffic system according to claim 4 is linearized and Learning Control Method, and it is special Levy and be, be the step of study control law and parameter more new law that dynamical linearization data model is set up in the step (5)：

(51) setting dynamical linearization data model meets and assumes 3, concurrently sets Expressway Traffic system in limited operation Between be spaced t in repeat,It is strict repetition；

Assuming that 3：PPD parameter matrixsIt is positive definite or nonnegative definite；

(52) set and expect that traffic is output as y_d,k(t)∈Rⁿ, for all of t ∈ { 0,1, ∧, T }, k=1,2, ∧, y_d,k(t) It is that iteration is related and bounded, i.e.,

Wherein, b_ydIt is normal number and presence；

(53) tracking error e is defined_k(t)=y_d,k(t)-y_k(t), then

e_k(t+1)=y_d,k(t+1)-y_k(t)-Φ(t)Δr_k(t)=Φ (t) (Φ (t)-¹y_d,k(t+1)-Φ(t)-¹y_k(t)-Δ r_k(t)) (9)

Order

e_k(t+1)=Φ (t) [Θ (t) ζ_k(t)-Δr_k(t)] (10)

Wherein, ζ_k(t)=y_d,k(t+1)-y_k(t)∈Rⁿ, Θ (t)=Φ (t)^-1∈R^n×n；

(54) the study control law that can then obtain kth time can be expressed as：

$r_k\left(t\right)=r_k(t-1)+{\widehat{\mathrm{\Θ}}}_k\left(t\right){\mathrm{\ζ}}_k\left(t\right),t=\{0,1,\mathrm{\Λ},T-1\}---\left(11\right)$

Wherein,It is the estimate of Θ (t), its parameter more new law is

${\widehat{\mathrm{\Θ}}}_k\left(t\right)={\widehat{\mathrm{\Θ}}}_{k-1}\left(t\right)+\frac{{\mathrm{aPe}}_{k-1}\left(t+1\right)}{c+{\mathrm{\ζ}}_{k-1}{\left(t\right)}^T{\mathrm{\ζ}}_{k-1}\left(t\right)}{\mathrm{\ζ}}_{k-1}{\left(t\right)}^T,---\left(12\right)$

Wherein,It is given bounded；C ＞ 0；0 ＜ ab_Φ＜ 2, P=I_n×nIt is unit matrix.